Dogs normally adjust to a new home quicker than cats. New neighbourhoods mean new smells and new animals. It can take time for your pet to feel at home here, so be patient.
For dogs, start by taking them on a on-lead walk around your new home and garden. This gives you a good opportunity to check for any holes in the fencing, or broken panels, and that it’s high enough for them not to jump over. Once you know the garden is fully secure, you can let them off for a proper explore.
If you haven’t moved that far, cats can pick up on familiar scents and head back to your old home. To help avoid this, give your cat plenty of time to get used to your new home - this should be about 2-3 weeks. Then, once they’re feeling confident and comfortable, you can let them outside while you are home to keep an eye on them. Start slowly, with short supervised visits to the garden (a cat harness can be useful). Time their garden visits for just before meal times. That way, you can tempt them back inside with food.
Ensure your pets know where to find the cat or dog door and are able to use it to get in and out if you are allowing them to free roam your backyard without supervision.